TITLE: Why is the power function considered an algebraic function, but the exponential function is NOT algebraic?
QUESTION [8 upvotes]: I am curious to know why if a mathematical expression contains an exponential function that expression may NOT be considered an algebraic expression, but if it contains a power function (if the variable is the base of a power expression) then that expression as a whole can be considered an algebraic expression.

REPLY [3 votes]: To add to the other answers, $x^\sqrt{2}$ where $x$ is a positive real is typically not considered algebraic because it is not obtainable by arithmetic and taking roots of polynomial equations. Note that $x^\sqrt{2} = \exp(\sqrt{2}\ln(x))$. So it is typically not true that "if the variable is the base of a power expression then that expression can be considered algebraic".